Problem Set
Sun Screen: A 'Pi in the Sky' Math Challenge
Overview
The "Pi in the Sky" math challenge gives students a chance to take part in recent discoveries and upcoming celestial events, all while using math and pi just like NASA scientists and engineers. In this problem from the set, students use the mathematical constant pi to find out how much solar energy is lost on Earth when Mercury transits, or passes in front of, the Sun.
Materials
Background
A transit happens when a planet crosses in front of a star. From our perspective on Earth, we only ever see two planets transit the sun: Mercury and Venus. Transits of Mercury happen about 13 times per century, while transits of Venus are even more rare. The next transit of Venus won't happen until 2117! As Mercury passes in front of the sun, viewers around Earth (using the proper safety equipment) will be able to see a tiny dark spot moving slowly across the disk of the sun.
Some objects, like the moon and Mercury, were originally thought to have no atmosphere. But scientists have discovered that these bodies are actually surrounded in an ultra-thin atmosphere of gases called an exosphere. Scientists want to better understand the composition and density of the gases that make up Mercury’s exosphere and transits make that possible.
“When Mercury is in front of the sun, we can study the exosphere close to the planet,” said NASA scientist Rosemary Killen. “Sodium in the exosphere absorbs and re-emits a yellow-orange color from sunlight, and by measuring that absorption, we can learn about the density of gas there.”
When Mercury transits the sun, it causes a slight dip in the sun’s brightness as it blocks a tiny portion of the sun's light. Scientists discovered they could use that phenomenon to search for planets orbiting distant stars, called exoplanets, that are otherwise obscured from view by the light of the star. When measuring the brightness of far-off stars, a slight recurring dip in the light curve (a graph of light intensity) could indicate an exoplanet orbiting and transiting its star. NASA’s Kepler mission has found more than 1,000 exoplanets by looking for this telltale drop in brightness.
Additionally, scientists have begun exploring the exospheres of exoplanets. By observing the spectra of the light that passes through an exosphere – similar to how we study Mercury’s exosphere – scientists are beginning to understand the evolution of exoplanet atmospheres as well as the influence of stellar wind and magnetic fields.
Procedures
A transit occurs when a planet passes in front of the disk of a star. As seen from Earth, only Mercury and Venus transit our star, the sun. During a transit, there is a slight dip in the amount of solar energy reaching Earth, which can be found using this equation:
B% = 100 (
)B = percentage drop in the brightness of the sun
r = the radius of the planet as it appears from Earth (in arcseconds)
R = the radius of the sun as it appears from Earth (in arcseconds)
With many solar-powered satellites orbiting Earth, it’s important to know what impact a dip in solar energy might have.
If 1,360.8 w/m2 of solar energy reaches the top of Earth’s atmosphere, how many fewer watts reach Earth when Mercury (diameter = 12 arcseconds) transits the sun (diameter = 1,909 arcseconds)?
Assessment
Extensions
Pi Day Challenges- Pi in the Sky
- Pi in the Sky 2
- Pi in the Sky 3
- Pi in the Sky 4
- Pi in the Sky Challenge (slideshow for students)
Multimedia
Features